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A theory is presented for Pierrehumbert's three-dimensional short-wave inviscid instability of the simple two-dimensional elliptical flow with velocity field u (x, y, z) = (-Ey, E^-1x, 0). The fundamental modes, which are also exact solutions of the nonlinear equations, are plane waves whose wave vector rotates elliptically around the z axis with period 2. The growth rates are the exponents of a matrix Floquet problem, and agree with those calculated by Pierrehumbert.
B. J. Bayly (Mon,) studied this question.